A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
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7 views

Square of a stationary process

I have to prove that if $X_t$ is a stationary process, $X_t^2$ is also. It is easy to prove the part referred to the means but I do not know how to work with covariances. Thanks!
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1answer
16 views

Is $X_t := W_t^2$ a Wiener process for a Wiener process $(W_t)_{t \geq 0}$?

I'm studying for exam and found this exercise which I don't really understand: Suppose $W_t$ is standard Wiener process. Is process $X_t=W_t^2, t\geq0$ a Wiener process? So I need to show that ...
2
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0answers
20 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
2
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0answers
26 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
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0answers
9 views

Higher dimensional SDEs cannot be compressed into a one dimensional SDE.

This question comes from quantitative finance but I think it's true in general outside that setting. I'm trying to make sense of the idea that if a process depends on at least two noises there ...
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1answer
26 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
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2answers
24 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
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1answer
30 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
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0answers
26 views

Independence of Poisson processes watched only some of the time

Let $(X_t)$ and $(Y_t)$ be independent homogeneous Poisson processes with rates $\lambda,\mu > 0$, and let $t_1, t_2, \dots$ and $t_1', t_2', \dots$ be two increasing sequences of possibly infinite ...
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1answer
13 views

how to solve for Ut stochastic question [on hold]

The process given by dUt = 􀀀-rUtdt + sigmadXt; U0 = u; where r,sigma are constants how to solve this equation for Ut? Thank you
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1answer
14 views

What are the conditions for $E[\int_0^tf(W_s,s)dW_s]=0$?

Let $W_t$ be the standard Brownian Motion. I am interested on the conditions on $f(\cdot)$ that guarantee that the expectation of the Ito integral below is zero: ...
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0answers
10 views

What is “starting” static distribution?

I'm not sure if I call everything correctly in English in here, but i have a problem with stochastic processes - Markov chains to be more specific. I'm calculating the "starting" stationary ...
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1answer
21 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
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1answer
18 views

Example of an adapted but not progressively measurable process

I'm looking for an example of a stochastic process $X$ that is $\mathbb{F}$-adapted, but not progressively measurable. One example I found is the following: $(\Omega, \mathfrak{A}) = (\mathbb{R^+}, ...
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0answers
16 views

Skorokhod vs Meyer zheng topology

I am new to the Skorokhod space and I want to know why Meyer-Zheng topology on the space of càdàg functions is weaker than the standard Skorokhod topology. Thanks in advance!
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0answers
7 views

Class properties Markov chain [on hold]

How can we show that an open class in a Markov chain is transient (both for finite and infinite)?
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1answer
16 views

Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration.

Let $\left(B_{t}\right)_{t\geq0}$ a Brownian motion. Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration. Remark: I try the following: It suffices to show ...
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0answers
7 views

No drift brownian motion and minimization at a given time [on hold]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ At a specific given time $ T = \tau $, how can I tell if ...
4
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1answer
51 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...
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1answer
29 views

What is the distribution of $B(t_1)+B(t_2)+…+B(t_n)$ [on hold]

$\{ B(t), t\ge 0\}$ is a standard Browian Motion Process. What is the distribution of $B(t_1)+B(t_2)+...+B(t_n)$ ?
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0answers
10 views

linear second moment of zero mean stochastic process with independent, stationary increments

I'm working on the following problem: Let $X$ be a zero mean stochastic process with independent and stationary increments. I want to prove that the function $t \mapsto \mathbb{E}X_t^2$ is linear. I ...
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1answer
23 views

counterexample to conditional expectation

Let F,G be some $\sigma-algebra$ is it true that in general $E\left(E\left(X\mid G\right)\mid F\right)\neq E\left(X\mid F\cap G\right)$? I think it's not, however I can't provide a counter example
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21 views

limit of sum of a brownian motion

Let $W_t$ be a wiener process and let $\pi$ be a partition of the segment $[0,T]:0\leq t_1\leq...\leq t_n=T$ I need to show without using the martingale property that the term below tends to $0$ in ...
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0answers
3 views

Limit distribution of absolute value maximum of stationary non-differentiable Gaussian process

Consider a real-valued stationary Gaussian Process $\{ X(t) \colon t \geq 0 \}$ with zero mean and unit variance and covariance function $r$ satisfying $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha}), ...
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1answer
91 views

Stationary Markov process properties

Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$, defined on $(\Omega, \mathcal{F}_t,P)$. Suppose that $X$ has stationary, independent increments. I now want to show the ...
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0answers
8 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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2answers
46 views

Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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4answers
55 views

A Stochastic Limiting Inequality Proof

Let $(X_p)_{p\ge 0}$ be a sequence of non-negative random variables with finite mean for each $p\ge 0$. Then $$\liminf_{p\to\infty} X_p^{\frac{1}{p}}\le \liminf_{p\to\infty}E(X_p)^{\frac{1}{p}}$$ ...
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0answers
15 views

why hull white model has normal distribution?

consider hull white model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha ...
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1answer
13 views

Inf is not a stopping time in general

If ${\tau_n}$ , $n=1,2,3...$ are stopping times to a given filtration $F_t$, why in general it's not true to claim that $\inf_n {\tau_n}$ is a stopping time also? Thanks
4
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1answer
54 views

A question related to reflection principle

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all ...
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1answer
20 views

Examples of predictable processes

I am asked to prove that the following processes are predictable. I am used to looking at stochastic processes as sequences of random variables (by fixing time) or as a collection of paths (by fixing ...
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0answers
13 views

Completion of a stochastic basis (Filtration)

Given a stochastic basis ($\Omega, \mathbb{F},(\mathbb{F_t})_{t \in \mathbb{R}},P$) with a right-continuous filtration, it is possible to construct a complete stochastic basis $\Omega, ...
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3answers
47 views

How to show $M_n = X_n^2-n$ is a martingale?

Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk. $X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$. Then how can I show that: A) $M_n = X_n^2-n$ is a ...
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0answers
40 views

Clarify a question's answer related to random walk. [closed]

I'm studying Problem5.3 and its solution. However, its solution is not clear for me. Please explanatorily show this answer . I need to learn such type of questions. Please help me. Thank you.
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1answer
60 views

A random walk question: what is the given probability?

Let $\{X_n\}_{n\in\Bbb N_0}$ be a simple random walk, given $n\in \Bbb N$ what is the probability $$ \mathbb P(X_1\ge0,X_2\ge0,\ldots, X_{2n-1}\ge0,X_{2n}=0) $$ I think that I should benefit from ...
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1answer
20 views

does brownian motion and poisson random measure have to be independent? [closed]

Suppose a brownian motion $W$ and a poisson random measure $\mu$ are defined on the same filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$, where both $W$ and $\mu$ are ...
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0answers
41 views

Distinct states of a Markov chain [closed]

I don't understand this problem,mostly part a. Can you explain me?
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1answer
52 views

Markov chain problem 13 [closed]

I have this problem I don't understand, Can you help me, please?
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1answer
27 views

Ehrenfest chain [closed]

I tried to find the solution, but I don't understand how to get it, because the book (Introduction to stochastic processes) says that the answer is $P(X_1=x)$=$P(X_0=x)$, but it doesn't make sense ...
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1answer
60 views

Every zero-mean Lévy process has linear variance (wrt $t$)

I'd like to show that every Lévy process with $\mathbb{E}X_t=0, \:\forall t\ge0$ has linear variance, namely $t\mapsto\mathbb{E}X^2_t$ is linear. I showed that indeed the additivity holds, i.e. ...
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1answer
35 views

About a probability space [closed]

Consider a probability space (Ω,A,P) and assume that the various sets mencioned below are all in A. (a) Show that if $D_i$ are disjoint and $P(C|Di)=p$ independently of i, then $P(C|⋃iDi)=p$. (b) ...
3
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1answer
33 views

Local maximum of brownian motions

Let $B=(B_t)_{t\geq 0}$ be the standard Brownian motion. I want to show that for every $t_0 \geq 0$ $\mathbb{P}$($B$ has a local maximum in $t_0$)=0. I've already shown that for every ...
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0answers
27 views

Stopping and optional times.

Let $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq0},P)$ be a filtered probability space. Put $\mathcal{F}_{t^+}:=\cap_{s>t}\mathcal{F}_s$ and $\{\mathcal{F}_{t^+}\}_{t\geq0}$ be the ...
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1answer
49 views

Concluding from limiting behavior

I've recently seen the following question on the internet: If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around? ...
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1answer
156 views

Markov process on an Abelian group

Let $E$ be an Abelian group. Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$ (where $\mathcal{E}$ denotes the $\sigma$-algebra on $E$), defined on $\Omega, \mathcal{F}_t,P)$. ...
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26 views

Absolute convergence of $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$

I want to show that if $X \in L^1$, where $X$ is a real-valued random variable, the sum $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$ converges absolute. My idea was the following: Since $X \in ...
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40 views

How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
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0answers
8 views

Markov Semigroups worked example

I have been reading this excellent paper on Markov semigroups, in which the assertion is made that a markov semigroup $\mathcal{P: L^1 \longrightarrow L^1}$ is defined by $\frac {d\mu}{dm}$ for some ...